From an entropic inequality for sums in abelian groups to the theorems of Ruzsa and Plünnecke

We give a direct information-theoretic proof of the Plünnecke–Ruzsa inequalities for finite subsets of abelian groups. The argument rests on an elementary coupling inequality proved by maximizing joint entropy, without tensor powers or asymptotic constructions. It also yields Ruzsa’s sumset triangle inequality and an extension to non-abelian groups.